View source for Lattice of points

''point lattice, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l0576602.png" />, with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l0576603.png" />''

The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l0576604.png" /> of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l0576605.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l0576606.png" /> are integers.

The lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l0576607.png" /> can be regarded as the free [[Abelian group|Abelian group]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l0576608.png" /> generators. A lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l0576609.png" /> has an infinite number of bases; their general form is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l05766010.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l05766011.png" /> runs through all integral matrices of determinant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l05766012.png" />. The quantity

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l05766013.png" /></td> </tr></table>

is the volume of the parallelopipedon formed by the vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l05766014.png" />. It does not depend on the choice of a basis and is called the determinant of the lattice <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l057/l057660/l05766015.png" />.

The partition of point lattices into [[Voronoi lattice types|Voronoi lattice types]] plays an important role in the geometry of quadratic forms (cf. [[Quadratic form|Quadratic form]]).



====Comments====
The idea of lattices and lattice points links geometry to arithmetic (integers). Therefore it plays a central role in the [[Geometry of numbers|geometry of numbers]]; [[Integer programming|integer programming]] (lattice point theorems); [[Diophantine approximations|Diophantine approximations]]; reduction theory; [[Analytic number theory|analytic number theory]]; numerical analysis; crystallography (cf. [[Crystallography, mathematical|Crystallography, mathematical]]); [[Coding and decoding|coding and decoding]]; [[Combinatorics|combinatorics]]; geometric algorithms, and other areas.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1972) {{MR|1434478}} {{MR|0306130}} {{MR|0181613}} {{MR|0157947}} {{ZBL|0866.11041}} {{ZBL|0209.34401}} {{ZBL|0131.29003}} {{ZBL|0086.26203}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989) {{MR|1003606}} {{ZBL|0683.10025}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint) {{MR|0893813}} {{ZBL|0611.10017}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> P.M. Gruber (ed.) J.M. Wills (ed.) , ''Handbook of convex geometry'' , North-Holland (1992) {{MR|1242973}} {{ZBL|0777.52002}} {{ZBL|0777.52001}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> R. Kannan, L. Lovasz, "Covering minima and lattice-point-free convex bodies" ''Ann. of Math.'' , '''128''' (1988) pp. 577–602 {{MR|0970611}} {{ZBL|0659.52004}} </TD></TR></table>